The compound interest on a certain sum of money at ` 5%` per annum for 2 years is `Rs. 246`. The simple interest on the same sum for 3 years at `6%` per annum is

Options

A. `Rs.435`
B. `Rs. 450 `
C. `Rs.430 `
D. `Rs. 432 `

Answer

To solve the problem step by step, we will first determine the principal amount using the compound interest formula, and then we will calculate the simple interest for the same principal over 3 years at a different interest rate.

### Step 1: Find the Principal Amount
We know that the compound interest (CI) for 2 years is Rs. 246 at a rate of 5% per annum.

The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) = Total amount after interest
- \(P\) = Principal amount
- \(r\) = Rate of interest
- \(n\) = Number of years

Here, \(r = 5\%\) and \(n = 2\). The compound interest can also be expressed as:
\[ CI = A - P \]
From the problem, we have:
\[ CI = 246 \]
Thus,
\[ A = P + 246 \]

Substituting \(A\) in the compound interest formula:
\[ P + 246 = P \left(1 + \frac{5}{100}\right)^2 \]
\[ P + 246 = P \left(1 + 0.05\right)^2 \]
\[ P + 246 = P \left(1.05\right)^2 \]
Calculating \(1.05^2\):
\[ 1.05^2 = 1.1025 \]
So, we have:
\[ P + 246 = 1.1025P \]
Rearranging gives:
\[ 246 = 1.1025P - P \]
\[ 246 = 0.1025P \]
Now, solving for \(P\):
\[ P = \frac{246}{0.1025} = 2400 \]

### Step 2: Calculate Simple Interest
Now that we have the principal amount \(P = 2400\), we will calculate the simple interest for 3 years at a rate of 6% per annum.

The formula for simple interest (SI) is:
\[ SI = \frac{P \times r \times t}{100} \]
Where:
- \(P = 2400\)
- \(r = 6\%\)
- \(t = 3\)

Substituting the values:
\[ SI = \frac{2400 \times 6 \times 3}{100} \]
Calculating:
\[ SI = \frac{2400 \times 18}{100} \]
\[ SI = \frac{43200}{100} = 432 \]

### Final Answer
The simple interest on the same sum for 3 years at 6% per annum is **Rs. 432**.

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